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(b)   If x, y e A are such that

\\c**y e-t*\\ ^ k\\y\\       (k a constant >0)

for all  6 C, show that xy = yx. (Use Liouville's theorem and the Taylor expansion
of the function  t-te^ye"** at  = 0.)

(c)    If \\x2\\ = ||je||2 for all x e A, show that A is commutative. (Observe that we have
p(x)= \\x\\ for all xeA, and apply this relation to e^ye'1**; then use (a) and (b).)

(d)    Show that, if there exists a constant k > 0 such that \\yx\\ <J k\\xy\\ for all x,y
in A, then A is commutative. (Apply the inequality with e~**x in place of x and e;*y
in place of y, and use (b).)

(e)    Let a e A be such that

for all  e C and all x e A. Prove that a belongs to the center of A. (Prove that, for
each f E C and all integers n^n0 (where n0 depends on )

for all y e A; then let n tends to + oo and use (b).)

15. Let A be a Banach algebra with unit element e, and let C be a commutative Banach
subalgebra containing e. Show that there exists a commutative Banach subalgebra
B of A containing C such that SpB(x) = SPA (x) for each x e C.


Let A be a commutative algebra (over C). A character of A is defined
to be any homomorphisrn x f the algebra A into C which is not identi-
cally zero. Because #(1*) = %x(x) for all scalars A, this condition is
equivalent to #(A) = C. If A has a unit element e ^ 0, then %(e) ^ 0
(otherwise x(x) =x(ex) = x(e)x(x)  0 for all x e A); it follows that x(e) = 1>
because %(e)2 =^(e2) = x(e)<

An ideal m ^ A in A is said to be maximal if there exists no ideal n such
that m T n, n ^ A and m c n. If A has a unit element, m is maximal if and
only if A/m is a (commutative) j?eW.

(15.3.1)   Let A be a commutative Banach algebra with unit element e = 0.
(i)   For each x e A and each character x of A, we have x(x) G SpA(:x;).

(ii)   Every character x of A is a continuous linear form with norm 1.

(iii) The mapping XI~>X~1(^) ^ a bijection of the set of characters of A
onto the set of maximal ideals of A (which are therefore closed).

Assertion (i) is a particular case of (15.2.8(i)). It follows (15.2.4(iii)) that
\x(x)\ ^ \\x\\, which shows (5.5.1) that ^ is a continuous linear form with
norm gl; and since %(e) = 1 and \\e\\ = 1, we have \\x\\ = 1. Finally, since(c).)